I have been trying to understand what a White Noise is and also a White noise Process and I have been trying to piece together different definitions that I have found online. I was wondering if my understanding of White Noise is correct
These notes say that the the mathematical definition of White Noise is and $0$ mean Generalized Gaussian Processes on $S$ (I believe that $S$ needs to be a smooth function of rapid decrease?), namely $Y(\phi)$, such that the variance of $Y(\phi)$ is $\int_{\mathbb R} \phi(t)^2 dt$.
So from my understanding, let $ \phi(t) \in S $ be a test function then we can consider the generalized stochastic process $$ X(\phi)=\int_{\mathbb R}\phi(t)dB_t $$ where $B_t$ is a brownian motion on $\mathbb R$. So $X$ here is the Wiener Integral. It is a standard result that $X(\phi)$ is a $0$-mean Gaussian random variable with variance $\int_{\mathbb R}\phi(t)^2dt $. So according to the above definition, $X$ is a white noise. But, in the same notes linked above, it says that White Noise can be thought of as the derivative of a Brownian Motion, which i'll refer to as $\dot{B}_t$.
Now using the stochastic integration by parts formula we see that $$ -\int_{\mathbb R}\phi '(t)B_tdt = \int_{\mathbb R}\phi(t)dB_t \tag{1} $$ and now we will informally denote this last integral as $$ \int_{\mathbb R}\phi(t)\dot{B}_tdt \tag{2} $$ So putting (1) and (2) together we see that $$ -\int_{\mathbb R}\phi '(t)B_tdt = \int_{\mathbb R}\phi(t)\dot{B}_tdt \tag{3} $$ And we now see that $\dot{B}_t$ is the weak or distributional dervitive of the Brownian motion $B_t$.
So now in the same notes listed above, the author says that the process $X(\phi)$, which was already said to be a White Noise, defines a White Noise $\dot{B}$. My question now is that does White Noise refer to both $X$ and $\dot{B}$? If we regard White Noise as the derivative (in the sense of distributions) of a Brownian Motion then $\dot{B}$ is a White Noise. But just using the mathematical definition of a White Noise we also see that $X$ is a White Noise.
Any input is appreciated!